Abstract: The estimation of locations of jump points and corresponding jump sizes of a density function on a bounded interval of interest by the kernel method is considered. Strong convergence rates (SCR) and limiting distributions for the proposed estimators are obtained. The order of the SCR for estimators of locations for jump points is immune to the smoothness conditions imposed on the density function, but that for estimators of jump sizes is not. The limiting distributions are used to test the continuity of the density function and give asymptotic confidence intervals for locations of jump points and corresponding jump sizes. For applications of these estimators, the choices of bandwidths and kernel functions are considered. In the case that the number of jump points on a bounded interval of interest is known in advance, an approach is proposed to recover the density function on the interval such that the performance of the resulting density function estimate is not affected by these jump points. Simulations demonstrate that the asymptotic results hold for reasonable sample sizes.
Key words and phrases: Asymptotic normality, density estimation, jump point, jump size, kernel estimator, strong consistency.